Transcutaneous organ function measurement

ABSTRACT

The present invention relates to a method for determining an organ function in a subject, comprising the steps of: providing a first concentration-time curve obtained by transcutaneously measuring in a body fluid at a first position background fluorescence in at least one first time point and fluorescence of an indicator compound in at least a second, a third, a fourth, a fifth, and a sixth time point; providing a second concentration curve obtained by transcutaneously measuring in a body fluid at a second position background fluorescence in at least one seventh time point and fluorescence of an indicator compound in at least a eighth, a ninth, a tenth, an eleventh, and a twelfth time point; fitting the first and the second concentration curve into a kinetic model representing at least four diffusion compartments; and thereby determining an organ function in a subject. The invention further relates to a device for determining an organ function according to the method of the present invention, said device comprising a first sensor for transcutaneously measuring fluorescence of an indicator at a first position, a second sensor for transcutaneously measuring fluorescence of an indicator at a second position; and a data processing unit for fitting the values obtained by the sensors into a kinetic model of one of the preceding claims. The present invention also relates to a kit comprising a device of the present invention and an indicator compound, as well as to a computer or computer network comprising at least one processor, wherein the computer or computer network is adapted to perform the method according to the present invention.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. 15/030,727,filed on Apr. 20, 2016, which is the U.S. national stage entry ofPCT/EP2014/072420, filed on Oct. 20, 2014, which claims priority toEuropean Patent Application 13189703.5, filed on Oct. 22, 2013, all ofwhich are incorporated by reference in their entirety.

The present invention relates to a method for determining an organfunction in a subject, comprising the steps of: providing a firstconcentration-time curve obtained by transcutaneously measuring in abody fluid at a first position background fluorescence in at least onefirst time point and fluorescence of an indicator compound in at least asecond, a third, a fourth, a fifth, and a sixth time point; providing asecond concentration curve obtained by transcutaneously measuring in abody fluid at a second position background fluorescence in at least oneseventh time point and fluorescence of an indicator compound in at leasta eighth, a ninth, a tenth, an eleventh, and a twelfth time point;fitting the first and the second concentration curve into a kineticmodel representing at least four diffusion compartments; and therebydetermining an organ function in a subject. The invention furtherrelates to a device for determining an organ function according to themethod of the present invention, said device comprising a first sensorfor transcutaneously measuring fluorescence of an indicator at a firstposition, a second sensor for transcutaneously measuring fluorescence ofan indicator at a second position; and a data processing unit forfitting the values obtained by the sensors into a kinetic model of oneof the preceding claims. The present invention also relates to a kitcomprising a device of the present invention and an indicator compound,as well as to a computer or computer network comprising at least oneprocessor, wherein the computer or computer network is adapted toperform the method according to the present invention.

In the clinical and preclinical field, determining various organfunctions is accorded great importance since, for example, correspondingtherapies or medications can be controlled in accordance with said organfunctions. The invention is described hereinafter substantially withregard to the kidney function. In principle, however, other applicationsare also conceivable in which the function of a particular organ can bedetected by means of the means and methods described herein. Thus, inprinciple, it is possible to use any desired exogenous or endogenoussubstances in the blood as indicator substances.

In kidney diagnostics, the quantitative and qualitative functionaltesting of the kidneys is of great significance. One indicator of thekidney function is the so-called glomerular filtration rate (GFR) whichindicates the amount of primary urine produced by the glomeruli of thekidneys per unit time.

For quantifying the GFR, several methods are known from the prior artand medical practice. One class of methods, into which the presentinvention may also be classified, is based on the use of one or aplurality of indicator substances, which are at least predominantlyremoved from the blood on account of the kidney function. This meansthat the indicator substance is removed from the body at leastpredominantly by the filtration effect of the glomeruli, in which casesubstantially neither tubular secretion nor resorption from the primaryurine takes place. Clearance is generally designated as that amount ofplasma in milliliters which is freed of the indicator substance perminute.

Various exogenous and/or endogenous indicator substances are known fordetermining the glomerular filtration rate. Examples of endogenousindicator substances are creatinine or cystatin C. Various exogenousindicator substances are also known from the prior art. Moreparticularly, saccharides, e.g. polyfructosans, can be used as indicatorsubstances. Examples of suitable indicator substances are disclosed inWO2001/85799 or WO2006/32441. It is generally possible to have recourseto this prior art in the context of the present invention as well.

From a metrological standpoint, one of the challenges consists, inparticular, in determining the concentration profile of the indicatorsubstance and thus the clearance thereof. Numerous different methods bymeans of which the clearance can be detected metrologically are compiledin WO 99/31183. Thus, some of the methods are based on the fact thatblood and/or urine samples are taken at regular or irregular intervals,and the concentration of the marker substance is determinedanalytically, for example by means of enzymatic detection methods. Othermethods are based on the use of radioactive indicator substances and/orX-ray contrast media. The acceptance of such indicator substances by thepatient is generally low, however. Methods based on determining therenal clearance by means of chemical or biochemical analysis or on theuse of radioactive indicator substances are generally complex andburdened with high errors. In routine clinical practice, therefore, inmany cases the kidney function is estimated on the basis ofapproximation formulae, which, however, are likewise very inaccurate andcan have error tolerances in the range of 30 to 40%.

The prior art therefore likewise discloses methods based on the use offluorescent markers. In this case, use is made of indicator substancesmarked with dyes that can be detected optically. By way of example,these can be fluorescent markers which are admixed with the indicatorsubstances or bonded to the indicator substances, for example bycovalent bonding. Examples of marked indicator substances are describedin WO2001/85799 or WO2006/32441, in which case it is possible to haverecourse to these marked indicator substances, for example, in thecontext of the present invention.

In the latter methods mentioned, therefore, an optical signal is used asa measure of the concentration of the indicator substance. In this case,the respective concentration of the indicator substance can be deducedfor example from a known relation between the optical signal and theconcentration. Said known relation can be, for example, of an empirical,semi- empirical or analytical nature, for example a relation determinedby means of calibration measurements. Thus, in DE 100 23 051 A1, forexample, the indicator substance used is sinistrin marked withfluoresceinisothiocyanate (FITC). In this case, a noninvasive,transcutaneous measurement of the FITC fluorescence signal by means of anoninvasive measuring head is described, inter alia. Said measuring headis configured as a fiber-optic measuring head in which an external lightsource, via an optical fiber, illuminates the skin and excites theFITC-sinistrin molecules contained therein. The fluorescent lightemitted by the FITC is in turn picked up by means of optical fibers andforwarded to an external detector.

However, the measurement of the fluorescence signals as described in DE100 23 051 A1 is extremely complex in terms of apparatus technology.This is because it is necessary to provide complex spectrographs inorder to evaluate the measurement signals. Moreover, a fiber-opticsystem is required which, on account of the associated losses ofexcitation light, necessitates the use of highly intensive lightsources, more particularly lasers. The fiber-optic system, together withthe complex light sources and lasers, has the effect, however, that ameasurement of the renal clearance cannot be carried out in an ambulantmanner or by means of portable equipment, but rather practicallyexclusively in optical laboratories specifically designed for thispurpose.

Similar indicator substances as described above have also been disclosedin WO2010/020673. However, in this document a simplified method ofmeasurement is described, using a portable sensor e.g. in the form of aplaster.

Numerous further analysis systems which, in principle, are also suitablefor portable equipment are generally known from other fields of medicaldiagnostics. Thus, US 2004/0210280 A1, for example, describes aplaster-like system which can be used for transdermal therapy anddiagnosis. Said document proposes, inter alia, that the systemindependently collects and takes up fluid samples from the skin. In A.Pais et al: High-sensitivity, disposable lab-on-a-chip with thin-filmorganic electronics for fluorescence detection, Lab Chip, 2008, 8,794-800, a disposable lab-on-a-chip test element is proposed. The latteris based on an organic light-emitting diode and an organicphotodetector. The test element is configured as a microfluidic testelement and is able to analyze liquid samples by means of fluorescencedetection. DE 10 2004 048 864 A1 describes an analytical test elementwith wireless data transmission which is used for determining theconcentration of an analyte from a body fluid. Said document proposesconfiguring at least a portion of the electrical components of thesystem on the basis of polymer electronics. US 2006/020216 A1 describesa portable health management apparatus that can be used, in particular,for a blood pressure measurement. Said document proposes, inter alia,measuring the movement of the blood within a blood vessel by means oflight absorption of light incident transdermally.

Generally, for kidney function testing in the prior art, recourse isregularly made to inulin as the gold standard. In this case, the inulinmeasurement is usually effected enzymatically in a serum or urinesample. Noninvasive methods using fluorescence-marked inulin yieldedambiguous results (WO2001/85799). FITC sinistrin was established as thestandard for fluorescence-based GFR determinations (WO2001/85799; Pill2005, Anal Bioanal Chem 382: 59-64; Pill 2005, Europ J Medicinal Chem40: 1056-1061), wherein here as well the measurements were predominantlyeffected in isolated samples.

However, these last-mentioned methods and devices known from the priorart either require sampling of blood and/or urine samples, whichdecreases acceptance by the patient, or they do not allow to directlydetermine the GFR or a parameter proportional thereto. For this reason,many methods determine plasma clearance of an indicator compound as anapproximation (reviewed in Frennby et al. (2002), Eur. Radiol 12: 475).

Consequently, one object of the present invention is to provide devicesand methods for determining organ functions, more particularly a kidneyfunction, which avoid the disadvantages of known devices and methods.More particularly, the intention is to provide a method of determiningthe GFR of a subject. This object is achieved by means of the inventionwith the features of the independent claims. Advantageous developmentsof the invention, which can be realized individually or in combination,are presented in the dependent claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows a graphical representation of a 2-compartment model forrenal function measurement.

FIG. 1B shows a fitting of an example measurement with the resulting2-compartment-model equation.

FIG. 2A shows a graphical representation of a 3-compartment model forrenal function measurement.

FIG. 2B shows a fitting of an example measurement with the resulting3-compartment-model equation.

FIG. 3A shows a graphical representation of a 4-compartment model forrenal function measurement.

FIG. 3B shows an example measurement with four simultaneously recordedconcentration-time curves.

FIG. 3C shows a measurement of bolus injection followed by constantinfusion until steady state is reached.

FIG. 4 shows an exemplary device according to the present invention,comprising a first sensor, a second sensor and a data processing unit.

DETAILED DESCRIPTION

Accordingly, the present invention relates to a method for determiningan organ function in a subject, comprising the steps of:

a) providing a first concentration-time curve obtained bytranscutaneously measuring in a body fluid at a first position (i)background fluorescence in at least one first time point and (ii)fluorescence of an indicator compound in at least a second, a third, afourth, a fifth, and a sixth time point,

b) providing a second concentration-time curve obtained bytranscutaneously measuring in a body fluid at a second position (i)background fluorescence in at least one seventh time point and (ii)fluorescence of an indicator compound in at least a eighth, a ninth, atenth, an eleventh, and a twelfth time point,

c) fitting the first and the second concentration-time curve into akinetic model representing at least four diffusion compartments, and

d) thereby determining an organ function in a subject.

As used in the following, the terms “have”, “comprise” or “include” orany arbitrary grammatical variations thereof are used in a non-exclusiveway. Thus, these terms may both refer to a situation in which, besidesthe feature introduced by these terms, no further features are presentin the entity described in this context and to a situation in which oneor more further features are present. As an example, the expressions “Ahas B”, “A comprises B” and “A includes B” may both refer to a situationin which, besides B, no other element is present in A (i.e. a situationin which a solely and exclusively consists of B) and to a situation inwhich, besides B, one or more further elements are present in entity A,such as element C, elements C and D or even further elements. From this,it is understood by the skilled person that the method of the presentinvention may, e.g., comprise obtaining more than two concentration-timecurves.

Further, as used in the following, the terms “preferably”, “morepreferably”, “most preferably”, “particularly”, “more particularly”,“specifically”, “more specifically” or similar terms are used inconjunction with optional features, without restricting alternativepossibilities. Thus, features introduced by these terms are optionalfeatures and are not intended to restrict the scope of the claims in anyway. The invention may, as the skilled person will recognize, beperformed by using alternative features. Similarly, features introducedby “in an embodiment of the invention” or similar expressions areintended to be optional features, without any restriction regardingalternative embodiments of the invention, without any restrictionsregarding the scope of the invention and without any restrictionregarding the possibility of combining the features introduced in suchway with other optional or non-optional features of the invention.

As used herein, the term “subject” relates to an animal, preferably amammal, more preferably a farm or companion animal. Most preferably, thesubject is a human.

The term “organ function”, as used herein, relates to the activity of anorgan of a subject of mediating and/or controlling passage of compoundsto or from blood. Thus, preferably, the term relates to brain-bloodbarrier function, intestinal wall barrier function, lung function, liverfunction, or pancreas function. More preferably, the term relates tokidney function. It is understood by the skilled person that the term“determining organ function”, as used herein, preferably relates todetermining a measurable parameter indicating, preferably beingproportional, to the function of the respective organ. It is alsounderstood by the skilled person that an organ may have more than onefunction and that distinct parameters may be measured in order todetermine distinct functions of an organ. Thus, preferably, determiningrenal function relates to determining the GFR.

As used herein, the term “providing a concentration-time curve” relatesto making available a concentration-time curve as detailed herein below.Preferably, providing a concentration-time curve relates to obtaining ameasuring curve according to the present invention. Thus, morepreferably, providing a first concentration-time curve is obtaining afirst concentration-time curve by transcutaneously measuring in a bodyfluid at a first position (i) background fluorescence in at least onefirst time point and (ii) fluorescence of an indicator compound in atleast a second, a third, a fourth, a fifth, and a sixth time point, andwherein providing a second concentration-time curve in step b) isobtaining a second concentration-time curve by transcutaneouslymeasuring in a body fluid at a second position (i) backgroundfluorescence in at least one seventh time point and (ii) fluorescence ofan indicator compound in at least a eighth, a ninth, a tenth, aneleventh, and a twelfth time point. As indicated above, for bothconcentration-time curves fluorescence is measured in at least one timepoint before administering the indicator substance to measure thebackground signal (baseline). It is understood by the skilled personthat the term may include further steps, e.g., preferably, contacting asubject to a measuring device, recording measurement values, and thelike.

The term “indicator compound”, as used herein, relates to a compoundemitting photons at an emission wavelength upon or after irradiationwith photons at at least one excitation wavelength. It is understood bythe skilled person that, preferably, the emission wavelength isnon-identical to the excitation wavelength. More preferably, theemission wavelength is shorter than the excitation wavelength(“up-conversion”). Most preferably, the emission wavelength is longerthan the excitation wavelength. In principle, compounds providing forall kinds of luminescence are suitable for the method of the presentinvention. However, compounds emitting light within a short time afterirradiation are preferred; thus, preferably the indicator compoundre-emits absorbed radiation on average within less than one s, morepreferably within less than one ms, even more preferably within lessthan a μs. Thus, most preferably, the indicator compound of the presentinvention is a fluorescent compound.

Preferably, the indicator compound of the present invention is anendogenous compound, i.e. a compound present in and/or produced in thebody of the subject. More preferably, the indicator compound is anexogenous compound, i.e. a compound not normally found in the body ofsaid subject. It is understood by the skilled person that an endogenouscompound may be administered to a subject in order to increase itsconcentration in the blood.

Preferably, the indicator compound comprises a fluorescent low-molecularweight compound covalently bound to a hydrophilic compound. Preferably,the indicator compound has a molar mass of less than 20 kg/mol, morepreferably less than 10 kg/mol, even more preferably less than 5 kg/mol,and most preferably less than 3 kg/mol. Preferably, the indicatorcompound is a zwitterionic or an acidic compound; more preferably, theindicator compound is a neutral compound, most preferably, the indicatorcompound is a basic compound. Preferably, the hydrophilic compound isselected from the list consisting of oligo- and polysaccharides, oligo-and polyalcohols, and oligo- and polyethers. Preferably, theoligosaccharide is a cyclodextrin, a fructan, or a dextran. Alsopreferably, the oligoalcohol is oligovinylalcohol and the polyalcohol ispolyvinylalcohol. Also preferably, the polyether is polyethylenglycol(PEG). Preferably, the indicator compound is a pharmacologicallyacceptable compound or can be provided in a pharmacologically acceptablepreparation.

The indicator compound is applied to the subject by any method suitableto ensure distribution in the blood of the subject. Thus, preferably,orally available indicator compounds may be used. More preferably, theindicator compound is administered to the blood stream of the subject,preferably intraarterially or intravenously. Preferably, the indicatorcompound is administered as a continuous infusion. More preferably, theindicator compound is administered as a bolus injection.

As used herein, the term “measuring fluorescence” relates to determiningfluorescence of the indicator compound comprised in a subjectsemiquantitatively or, preferably, quantitatively. Methods of measuringfluorescence are well-known in the art and generally include irradiatingan indicator compound with photons having an excitation wavelength anddetecting photons emitted by the indicator compound as a result ofexcitation irradiation. Suitable means for irradiating and detecting aredescribed herein below. Preferably, measuring fluorescence is achievedby one of the sensory devices described in WO2010/020673.

Preferably, fluorescence at a first position is measured at at least sixnon-identical points in time, i.e. at a first, a second, a third, afourth, a fifth, and a sixth point in time, wherein at the first pointin time, background fluorescence is determined. Thus, a first data setis provided consisting of at least six data pairs of a point in time,each correlated with a value of fluorescence of the indicator compound.It is understood by the skilled person that said first data setcorresponds to the change of concentration of the indicator compound inthe body fluids relevant for measurement at said first position. Thus,this data set can be used to establish a concentration-time curvecorresponding to the concentration change of the indicator compound inthe tissue under said first position. Preferably, the value measured asbackground fluorescence at the first point in time is subtracted fromthe values measured at the second to sixth points in time to providecorrected measurement values.

Also preferably, fluorescence at a second position is measured at atleast six non-identical points in time, i.e. at a seventh, an eighth, aninth, a tenth, an eleventh, and a twelfth point in time. Thus, a seconddata set is provided consisting of at least six data pairs of a point intime, each correlated with a value of fluorescence of the indicatorcompound. It is understood by the skilled person that said second dataset corresponds to the change of concentration of the indicator compoundin the body fluids relevant for measurement at said second position.Thus, this second data set can be used to establish a concentration-timecurve corresponding to the concentration change of the indicatorcompound in the tissue under said second position. Preferably, the valuemeasured as background fluorescence at the seventh point in time issubtracted from the values measured at the second to sixth points intime to provide corrected measurement values.

Preferably, the time frame encompassed by the first to sixth points intime overlaps with the time frame encompassed by the seventh to twelfthpoint in time. More preferably, said time frames overlap by at least 50%of the duration of the shorter time frame. Most preferably, said timeframes are the same. Preferably, the first to sixth and the seventh totwelfth point in time are evenly spaced over the respective time frameencompassed. Preferably, measuring at the at least two positions isperformed alternatingly, i.e., a measurement at a first position isfollowed by a measurement at a second position and vice versa. Morepreferably, measuring fluorescence of an indicator compound at a firstand a second position is done simultaneously, i.e. measuring at thefirst and the seventh point in time is performed at the same point intime, i.e. simultaneously, measuring at the second and the eighth pointin time is performed at the same point in time, and the like.

Preferably, the points in time of measuring are selected such that thefirst and seventh point in time lie before application of the indicatorcompound, such that at least one of the second to fifth and of theeighth to eleventh points in time lies such that the concentration ofthe indicator compound in blood is increasing with time, and such thatat least one of the third to sixth and of the ninth to twelfth points intime lies such that the concentration of the indicator compound in bloodis decreasing with time. As will be appreciated by the skilled person,such selection of time point is preferably achieved by obtainingmeasurement values at regular intervals and by selecting the relevantpoints in time as described above afterwards. Preferably, the selectionof time points as described above is assisted by an appropriatealgorithm. It is understood by the skilled person that the concentrationof the indicator compound may be unchanged or essentially unchangedbetween two adjacent points in time, e.g. in case the GFR is measuredand the subject is afflicted with renal failure. Thus, most preferably,the points in time are selected such that the concentration of theindicator compound is first increasing then decreasing with time in ahealthy reference subject.

Thus, in case the organ function is renal function or the GFR,preferably, the first and seventh point in time of measurement areselected 0 to 1 hour before administration of the indicator compound andthe later points in time of measurement are selected in the range of 0to 24 hours after administration of the indicator compound. Morepreferably, the first and seventh point in time of measurement areselected 0 to 0.5 hour before administration of the indicator compoundand the later points in time of measurement are selected in the range of0 to 12 hours after administration of the indicator compound. Mostpreferably, the first and seventh point in time of measurement areselected 0 to 0.25 hour before administration of the indicator compoundand the later points in time of measurement are elected in the range of0 to 6 hours after administration of the indicator compound. Preferably,in such a case all points in time of measuring are within 8 hours, morepreferably within 6 hours, even more preferably within 4 hours, or, mostpreferably, within 2 hours. It is, however, understood by the skilledperson, that especially in the case of impaired organ function, longermeasurement intervals may be required and that in such case themeasurement period may extend beyond 8 hours. Thus, in a preferredembodiment, the method of the present invention comprises a step offirst determining the concentration of the indicator compound at twonon- identical points in time, and, based on the result of saidmeasurements, deciding on the interval of measurement.

The term “concentration-time curve” is understood by the skilled personand, preferably, relates to a mathematical correlation of the change inconcentration of an indicator compound in a body fluid of a subject withtime. Thus, the concentration-time curve may comprise a plurality ofconcentration values acquired at different points in time. Theconcentration values may be acquired at a constant rate and/or atpredetermined or known points in time. Specifically in case a constantacquisition rate is used, the measurement curve may simply contain asequence of concentration values. The measurement curve, additionally,may comprise time information for each concentration values. Thus, as anexample, the measurement curve may contain value pairs, each value paircomprising a concentration value and time information regardingacquisition time of the respective concentration value. Accordingly, aconcentration-time curve is, preferably, obtained by determining atleast six concentration values at non-identical points in time andcorrelating the time/concentration pairs thus obtained mathematically,e.g. in a concentration over time graph or in a database. Since indilute solutions the fluorescence of an indicator compound isproportional to the concentration of said indicator compound, theskilled person understands that a concentration- time curve may also berepresented by a mathematical correlation of the change in fluorescenceof an indicator compound in a body fluid of a subject with time.

The terms “transcutaneous” and “transcutaneously”, as used herein,relate to a mode of measurement wherein at least one of providingphotons for illuminating the indicator compound present in the blood,preferably the interstitial fluid of a subject, and detecting lightemitted by said indicator compound is performed on the body surface of asubject. Preferably, both providing photons for illuminating theindicator compound and detecting light emitted by the indicator compoundare performed on the body surface of a subject. The term “body surface”relates to all outer or inner surfaces of a body connected to thesurrounding environment. Thus, preferably, the body surface of thepresent invention is a part of the body accessible without performing asurgical measure like, e.g. incision, or cutting the skin. Preferably,the body surface is mucosa or a skin, more preferably a non- ormoderately hirsute section of the skin. Preferably, the body surface isthe skin of a finger or toe, of an arm or leg, or of the torso, or afingernail or a toenail.

The terms “interstitial fluid” and “interstitium”, as used herein,relate to the intrabodily liquid surrounding cells of a subject outsideof blood vessels. As appreciated by the skilled person, the termpreferably excludes fluid within epithelial lined body cavities, like,e.g. cerebrospinal fluid, joint fluid, and bladder urine.

The term “position”, as relating to measurement at a specific position,relates to a point on the body surface of a subject. Preferably, theposition is selected as to enable reliable and, more preferably,unobstructive measurement. According to the present specification, thefirst position and the second position are non-identical, meaning thatat least one of the light path between light source and indicatorcompound molecules and the light path between indicator compoundmolecules and the detector are non-identical for said two positions.More preferably, both the light path between light source and indicatorcompound molecules and the light path between indicator compoundmolecules and the detector are non-identical for said two positions. Ina preferred embodiment, the first position is different from the secondposition, meaning that the first position is separated by at least 0.001m from the second position; more preferably the first position isseparated by at least 0.005 m from the second position, even morepreferably by at least 0.01 m, most preferably by at least 0.1 m.

The term “kinetic model”, as used herein, relates to a kinetic model ofconcentration change of a compound in a diffusion compartment. Suchmodels are, in principle, known from the prior art and are discussed indetail in the Examples of the present invention.

The same mathematical basis can be applied to models describing two- ormore compartment models. Preferably, the kinetic model of the presentinvention is a three-compartment model. Preferably, in case the organfunction is renal function, the four-compartment model is based on thefollowing assumptions: (i) the four compartments relevant fordistribution of the indicator compound are plasma, interstitial fluid,the local interstitium at the position of measurement, and adistribution compartment, corresponding to the container containing theindicator compound before application to the body of the subject; (ii)the indicator compound is eliminated only from plasma, not from theother compartments; (iii) the indicator compound is distributed from thedistribution compartment into plasma (iv) the indicator compound isdistributed between plasma and interstitial fluid in both directions;(v) the concentration of the indicator compound in the distributioncompartment at time 0 is maximal; (vi) the time required fordistribution of the indicator compound from the distribution compartmentis >0, meaning that the plasma concentration of the indicator compoundat the time of application is not the maximal concentration obtainable;(vii) the local interstitium exchanges indicator compound with theplasma compartment, and (viii) exchange with the small volume of thelocal interstitium does not change the concentration of the indicatorcompound in plasma, or the interstitial fluid, respectively.

Preferably, in case the organ function is renal function, theconcentration-time curves obtained are fitted into said four compartmentmodel. Preferably, a result of said fitting is the GFR per plasma volumeof the subject. More preferably, the GFR is calculated from theaforementioned result by multiplication of said result with the plasmavolume of the subject. It is understood by the skilled person that theaforementioned calculation of the GFR from the result of the curve fitmay also be included as a further calculation step after fitting theconcentration-time curves, such that the GFR is issued directly as theresult. Preferably, fitting is performed by processing the dataunderlying the concentration-time curves with a data processing unit,e.g., preferably, a computer or computer network.

It is understood by the skilled person that, based on the aboveassumptions, a kinetic model can be established and that theconcentration-time curves obtained by measuring the concentration at afirst and a second position at at least six points in time,respectively, can be fitted into the aforementioned kinetic model, asdisclosed herein in the Examples. Preferably, fitting is performed byestimating the optimal values of the relevant parameters, e.g. GFR/VP,the exponentials lambda 1 . . . lambda3, R3,L/VI,L using a suitablemathematical method, e.g. Reduced Chi-Square. Preferably, fitting isperformed by processing the data underlying the concentration-timecurves with a data processing unit, e.g., preferably, a computer orcomputer network.

Preferably, the plasma volume of a subject is determined by one of themethods known to the skilled person as explained e.g. in Margouleff(2013) Clin Nucl Med. 38(7): 534-7 or in Wang et al. (2010) Am J PhysiolRenal Physiol. 299(5): F1048-55.More preferably, the plasma volume ofthe subject is determined from the body mass of the subject according toempirical data available e.g. from Probst et al. (2006), Journal of theAmerican Association for Laboratory Animal Science: JAALAS, 45(2): 49.

In another preferred embodiment, fitting the first and the secondconcentration-time curve into a kinetic model representing at least fourdiffusion compartments comprises fitting said first and secondconcentration-time curve into a three-compartment model followed byincorporating a fourth compartment into the results of said fitting.

Preferably, the method of the present invention is modified orcomplemented by one or more of the following further steps:

Preferably, before obtaining a concentration-time curve, fluorescence ofan indicator compound in a body fluid is measured at at least two pointsin time and it is determined whether said two fluorescence values differsignificantly, and the intervals between the first to fifth and thesixth to tenth points in time are adjusted depending on the result ofsaid determination.

Also preferably, data sets for concentration-time curves are obtainedcontinuously and a quality control algorithm is applied to the fittingof the concentration-time curves, and a signal is provided when the datapermit determination of an organ function within a predefinedsignificance range. It is understood by the skilled person that saidsignal may e.g., preferably, be termination of measuring fluorescence.

Advantageously, it was found in the work underlying the presentinvention that an organ function, preferably renal function, morepreferably the GFR, can be determined by measuring the change ofconcentration of an indicator compound in a subject transcutaneously. Inparticular, it was found that by applying suitable mathematical modelsand by recording two concentration-time curves at two distinct positionson the skin of a subject, local parameters can be eliminated from thecalculations, such that as a direct result of curve fitting the GFR perplasma volume of the patient is obtained. Since the method of thepresent invention is also applicable in cases where the marker isapplied in a bolus injection, the GFR can be determined without anyfurther incision of the body surface of the subject. Where suitableorally applicable markers are used, the whole determination can beaccomplished non-invasively.

The definitions made above apply mutatis mutandis to the following.Additional definitions and explanations made further below also applyfor all embodiments described in this specification mutatis mutandis.

The present invention also relates to a device for determining an organfunction according to the method of the present invention, said devicecomprising a) a first sensor for transcutaneously measuring fluorescenceof an indicator at a first position, b) a second sensor fortranscutaneously measuring fluorescence of an indicator at a secondposition; and c) a data processing unit for fitting the values obtainedby the sensors into a kinetic model according to the present invention.

The term “device”, as used herein, relates to a system of meanscomprising at least the aforementioned means operatively linked to eachother as to allow the determination. Preferred means fortranscutaneously measuring fluorescence of an indicator compound, andmeans for carrying out the fitting are disclosed herein above inconnection with the method of the invention and below in connection withdisclosure related to data processing. How to link the means in anoperating manner will depend on the type of means included into thedevice. For example, where means for transcutaneously measuringfluorescence of an indicator compound are applied, the data obtained bysaid transcutaneously measuring fluorescence of an indicator compoundcan be processed by, e.g., a computer program in order to obtain thedesired results. Preferably, the means are comprised by a single devicein such a case. Said device may accordingly include two sensor units fortranscutaneously measuring fluorescence of an indicator compound and acomputer unit for processing the resulting data for the evaluation.

As used herein, the term “sensor” relates to a device or part thereofenabling the detection of photons emitted by an indicator compound uponirradiation with an excitation light. Preferably, the sensor unitadditionally comprises a source of excitation light. The person skilledin the art will realize how to link the means without further ado.Preferred sensors are disclosed e.g. in WO2010/020673, e.g. a sensor inthe form of a sensor plaster. Preferably, two sensors, e.g. sensorplasters, are operatively linked in that the data processing unit of thesystem brings together the results of the measurements performed by thesensors and fits the data into the kinetic model. The sensors may appearas separate devices in such an embodiment and are, preferably, packagedtogether as a kit. It is, however, also envisaged that two sensor unitsmay be comprised in one casing, preferably in cases where the distancebetween the first and the second measurement position is short. Inanother preferred embodiment, one source of excitation light is arrangedin such a way that the indicator compound at two different measurementpositions is irradiated. Preferred devices are those which can beapplied without the particular knowledge of a specialized clinician. Theresults may be given as output of raw data which need interpretation bythe clinician. Preferably, the output of the device is, however,processed, i.e. evaluated, raw data the interpretation of which does notrequire a clinician.

Preferably, the device further comprises a data storage unit comprisingat least reference values of total plasma volume correlated to bodymass. As described herein above, in case the organ function is the GFRand in case the body mass of the subject is provided, the device maydirectly calculate the GFR from the result of fitting theconcentration-time curves to the kinetic model in combination with theplasma volume estimate derived from the body mass of said subject.

Moreover, the present invention relates to a kit comprising a device ofthe present invention and an indicator compound.

The term “kit”, as used herein, refers to a collection of theaforementioned compounds, means or reagents of the present inventionwhich may or may not be packaged together. The components of the kit maybe comprised by separate units (i.e. as a kit of separate parts).Moreover, it is to be understood that the kit of the present inventionis to be used for practicing the method referred to herein above. It is,preferably, envisaged that all components are provided in a ready-to-usemanner for practicing the method referred to above. Further, the kitpreferably contains instructions for carrying out said method. Theinstructions can be provided by a user's manual in paper or electronicform. Preferably, the manual comprises instructions for interpreting theresults obtained when carrying out the aforementioned method using thekit of the present invention. Alternatively or in addition, the manualpreferably comprises reference values of plasma volume in dependence ofthe body mass of a subject.

Also, the present invention relates to a computer program, wherein thecomputer program is adapted to perform the method according to one ofthe preceding claims relating to a method while the program is beingexecuted on a computer.

Thus, the invention further discloses and proposes a computer programincluding computer- executable instructions for performing the methodaccording to the present invention in one or more of the embodimentsenclosed herein when the program is executed on a computer or computernetwork. Specifically, the computer program may be stored on acomputer-readable data carrier. Thus, specifically, one, more than oneor even all of method steps a) to d) as indicated above may be performedor assisted by using a computer or a computer network, preferably byusing a computer program. Specifically, method steps c) and d) may fullyor partially be performed by using a computer or computer network.Method steps a) and/or b) may fully or partially be embodied by using acomputer or a computer network, such as by providing the first andsecond concentration-time curves by the computer or computer network.Thus, generally, “obtaining” the first and second measurement curves mayalso imply providing the measurement curves, such as by providing themeasurement curves as stored in a data memory or data base, forprocessing in steps c) and d).

The invention further discloses and proposes a computer program producthaving program code means, in order to perform the method according tothe present invention in one or more of the embodiments enclosed hereinwhen the program is executed on a computer or computer network.Specifically, the program code means may be stored on acomputer-readable data carrier.

Further, the invention discloses and proposes a data carrier having adata structure stored thereon, which, after loading into a computer orcomputer network, such as into a working memory or main memory of thecomputer or computer network, may execute the method according to one ormore of the embodiments disclosed herein.

The invention further proposes and discloses a computer program productwith program code means stored on a machine-readable carrier, in orderto perform the method according to one or more of the embodimentsdisclosed herein, when the program is executed on a computer or computernetwork. As used herein, a computer program product refers to theprogram as a tradable product. The product may generally exist in anarbitrary format, such as in a paper format, or on a computer-readabledata carrier. Specifically, the computer program product may bedistributed over a data network.

Finally, the invention proposes and discloses a modulated data signalwhich contains instructions readable by a computer system or computernetwork, for performing the method according to one or more of theembodiments disclosed herein.

Preferably, referring to the computer-implemented aspects of theinvention, one or more of the method steps or even all of the methodsteps of the method according to one or more of the embodimentsdisclosed herein may be performed or assisted by using a computer orcomputer network. Thus, generally, any of the method steps includingprovision and/or manipulation of data may be performed by using acomputer or computer network. Generally, these method steps may includeany of the method steps, typically except for method steps requiringmanual work, e.g. certain aspects of performing the actual measurements.

Specifically, the present invention further discloses:

-   -   A computer or computer network comprising at least one        processor, wherein the processor is adapted to perform the        method according to one of the embodiments described in this        description,    -   a computer loadable data structure that is adapted to perform        the method according to one of the embodiments described in this        description while the data structure is being executed on a        computer,    -   a computer program, wherein the computer program is adapted to        perform the method according to one of the embodiments described        in this description while the program is being executed on a        computer,    -   a computer program comprising program means for performing the        method according to one of the embodiments described in this        description while the computer program is being executed on a        computer or on a computer network,    -   a computer program comprising program means according to the        preceding embodiment, wherein the program means are stored on a        storage medium readable to a computer,    -   a storage medium, wherein a data structure is stored on the        storage medium and wherein the data structure is adapted to        perform the method according to one of the embodiments described        in this description after having been loaded into a main and/or        working storage of a computer or of a computer network, and    -   a computer program product having program code means, wherein        the program code means can be stored or are stored on a storage        medium, for performing the method according to one of the        embodiments described in this description, if the program code        means are executed on a computer or on a computer network.

All references cited in this specification are herewith incorporated byreference with respect to their entire disclosure content and thedisclosure content specifically mentioned in this specification.

Figure Legends

FIG. 1:

A: A graphical representation of the 2-compartment model for renalfunction measurement.

B: Fitting of an example measurement with the resulting2-compartment-model equation.

FIG. 2:

A: A graphical representation of the 3-compartment model for renalfunction measurement.

B: Fitting of an example measurement with the resulting3-compartment-model equation.

FIG. 3:

A: A graphical representation of the 4-compartment model for renalfunction measurement.

B: Example measurement with four simultaneously recordedconcentration-time curves.

C: Measurement of bolus injection followed by constant infusion untilsteady state is reached.

FIG. 4: An exemplary device according to the present invention,comprising a first sensor 110, a second sensor 120 and a data processingunit 130.

LIST OF REFERENCE SYMBOLS

-   t time in minutes-   nsig normalized signal-   D device-   H distribution compartment-   P plasma compartment-   I interstitial compartment-   I,L local interstitial compartment-   R diffusion rate x interfacial area-   M measured data points-   FC fitted curve-   110 first sensor-   120 second sensor-   130 data processing unit

The following Examples shall merely illustrate the invention. They shallnot be construed, whatsoever, to limit the scope of the invention.

EXAMPLE 1 Two-Compartment Model

A graphical representation of the two-compartment model for renalfunction measurement known in the art is shown in FIG. 1A.

Example 2 Three-Compartment Model

A graphical representation of the three-compartment model for renalfunction measurement is shown in FIG. 2A.

Kinetic Model

As shown in E. L. Cussler. Diffusion: Mass transfer in fluid systems.Cambridge university press, 1997:

$\begin{pmatrix}{{amount}\mspace{14mu} {of}\mspace{14mu} {mass}} \\{transferred}\end{pmatrix} = {{k\begin{pmatrix}{interfacial} \\{area}\end{pmatrix}}\begin{pmatrix}{concentration} \\{difference}\end{pmatrix}}$

Since the interfacial area is unknown, it is summarized:

$R = {k\begin{pmatrix}{interfacial} \\{area}\end{pmatrix}}$

The mass changes over time in the model of FIG. 2A can be representedas:

$\begin{matrix}{\frac{{dm}_{H}}{dt} = {{- R_{4}} \cdot c_{H}}} & (0.1) \\{\frac{{dm}_{P}}{dt} = {{R_{4} \cdot c_{H}} - {R_{1} \cdot c_{P}} - {R_{2} \cdot c_{P}} + {R_{3} \cdot c_{I}}}} & (0.2) \\{\frac{{dm}_{I}}{dt} = {{R_{2} \cdot c_{P}} - {R_{3} \cdot c_{I}}}} & (0.3)\end{matrix}$

Since the signal is proportional not to the mass m_(I) but theconcentration c_(I) in the interstitium, the equations 0.1 to 0.3 needto be transformed with eq. 0.5:

$\begin{matrix}{c = \frac{m}{V}} & (0.4) \\{\frac{dc}{dt} = {\frac{1}{V}\frac{dm}{dt}}} & (0.5)\end{matrix}$

With V: Volume

$\begin{matrix}{\frac{{dc}_{H}}{dt} = {{- c_{H}} \cdot \frac{R_{4}}{V_{H}}}} & (0.6) \\{\frac{{dc}_{P}}{dt} = {{c_{H} \cdot \frac{R_{4}}{V_{P}}} - {c_{P} \cdot \frac{R_{1} + R_{2}}{V_{P}}} + {c_{I} \cdot \frac{R_{3}}{V_{P}}}}} & (0.7) \\{\frac{{dc}_{I}}{dt} = {{c_{P}\frac{R_{2}}{V_{I}}} - {c_{I} \cdot \frac{R_{3}}{V_{I}}}}} & (0.8)\end{matrix}$

Equation 0.6 can be solved directly:

$\begin{matrix}{c_{H} = {c_{H,\max} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}} & (0.9)\end{matrix}$

from eq. 0.8:

$\begin{matrix}{c_{P} = {{\frac{V_{I}}{R_{2}} \cdot \frac{{dc}_{I}}{dt}} + {\frac{R_{3}}{R_{2}} \cdot c_{I}}}} & (0.10) \\{\frac{{dc}_{P}}{dt} = {{\frac{V_{I}}{R_{2}} \cdot \frac{d^{2}c_{I}}{{dt}^{2}}} + {\frac{R_{3}}{R_{2}} \cdot \frac{{dc}_{I}}{dt}}}} & (0.11)\end{matrix}$

Using eq. 0.10, 0.9 and 0.11 in eq. 0.7:

$\begin{matrix}{{{\frac{V_{I}}{R_{2}} \cdot \frac{d^{2}c_{I}}{{dt}^{2}}} + {\frac{R_{3}}{R_{2}} \cdot \frac{{dc}_{I}}{dt}}} = {{\frac{R_{4}}{V_{P}} \cdot c_{H,\max} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}} - {\frac{R_{1} + R_{2}}{V_{P}} \cdot \left( {{\frac{V_{I}}{R_{2}} \cdot \frac{{dc}_{I}}{dt}} + {\frac{R_{3}}{R_{2}} \cdot c_{I}}} \right)} + {\frac{R_{3}}{V_{P}} \cdot c_{I}}}} & (0.12) \\{{\frac{d^{2}c_{I}}{{dt}^{2}} + {\frac{{dc}_{I}}{dt} \cdot \left( \frac{{R_{1} \cdot V_{I}} + {R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}}{V_{P} \cdot V_{I}} \right)} + {c_{I} \cdot \left( \frac{R_{1} \cdot R_{3}}{V_{P} \cdot V_{I}} \right)}} = {\frac{R_{2} \cdot R_{4}}{V_{P} \cdot V_{I}} \cdot c_{H,\max} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}} & \;\end{matrix}$

Equation 0.12 is an inhomogeneous differential equation of 2nd orderwith constant coefficients.

Homogeneous Solution

$\begin{matrix}{\mspace{76mu} {{{y_{h}^{''} + {y_{h}^{\prime} \cdot a} + {y_{h} \cdot b}} = 0}{{\frac{d^{2}y_{h}}{{dt}^{2}} + {\frac{{dy}_{h}}{dt} \cdot \left( \frac{{R_{1} \cdot V_{I}} + {R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}}{V_{P} \cdot V_{I}} \right)} + {y_{h} \cdot \left( \frac{R_{1} \cdot R_{3}}{V_{P} \cdot V_{I}} \right)}} = 0}}} & (0.13)\end{matrix}$

The solution of 0.13 is:

$\begin{matrix}{\mspace{79mu} {y_{h} = {{C_{1} \cdot e^{\lambda_{1} \cdot t}} + {C_{2} \cdot e^{\lambda_{2} \cdot t}}}}} & (0.14) \\{\mspace{79mu} {with}} & \; \\{\mspace{79mu} {\lambda_{1,2} = {{- \frac{a}{2}} \pm \sqrt{\frac{a^{2}}{4} - b}}}} & \; \\{\lambda_{1,2} = {{- \frac{{R_{1} \cdot V_{I} \cdot R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}}{2 \cdot V_{P} \cdot V_{I}}} \pm \sqrt{\frac{\left( {{R_{1} \cdot V_{I}} + {R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}} \right)^{2}}{4 \cdot V_{P}^{2} \cdot V_{I}^{2}} - \frac{R_{1} \cdot R_{3}}{V_{P} \cdot V_{I}}}}} & \;\end{matrix}$

Inhomogeneous Solution

$\begin{matrix}{\mspace{79mu} {{g(x)} = {\frac{R_{2} \cdot R_{4}}{V_{P} \cdot V_{I}} \cdot c_{H,\max} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}}} & (0.15) \\{\mspace{79mu} {y_{p} = {C_{3} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}}} & (0.16) \\{\mspace{79mu} {y_{p}^{\prime} = {{- \frac{R_{4}}{V_{H}}} \cdot C_{3} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}}} & (0.17) \\{\mspace{79mu} {y_{p}^{''} = {\frac{R_{4}^{2}}{V_{H}^{2}} \cdot C_{3} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}}} & (0.18) \\{\mspace{79mu} {{y_{p}^{''} + {a \cdot y_{p}^{\prime}} + {b \cdot y_{p}}} = {g(x)}}} & \; \\{{{\frac{R_{4}^{2}}{V_{H}^{2}} \cdot C_{3} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}} - {\frac{{R_{1} \cdot V_{I}} + {R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}}{V_{P} \cdot V_{I}} \cdot \frac{R_{4}}{V_{H}} \cdot C_{3} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}} + {\frac{R_{1} \cdot R_{3}}{V_{P} \cdot V_{I}} \cdot C_{3} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}} = {\frac{R_{2} \cdot R_{4}}{V_{P} \cdot V_{I}} \cdot c_{H,\max} \cdot e^{{- \frac{R_{4}}{V_{H}}} \cdot t}}} & (0.19) \\{\mspace{79mu} {C_{3} = {- \frac{R_{2} \cdot R_{4} \cdot c_{H,\max}}{\begin{matrix}{{R_{1} \cdot R_{4} \cdot \frac{V_{I}}{V_{H}}} + {R_{2} \cdot R_{4} \cdot \frac{V_{I}}{V_{H}}} +} \\{{R_{3} \cdot R_{4} \cdot \frac{V_{P}}{V_{H}}} - {R_{4}^{2} \cdot \frac{V_{P} \cdot V_{I}}{V_{H}^{2}}} - {R_{1} \cdot R_{3}}}\end{matrix}}}}} & (0.20)\end{matrix}$

General Solution—Linear Combination of the Homogenous and InhomogeneousSolution

$\begin{matrix}{{c_{I}(t)} = {y_{h} + y_{p}}} & (0.21) \\{{c_{I}(t)} = {{C_{1} \cdot e^{\lambda_{1} \cdot t}} + {C_{2} \cdot e^{\lambda_{2} \cdot t}} + {C_{3} \cdot e^{\lambda_{3} \cdot t}}}} & (0.22) \\{with} & \; \\{{c_{I}(0)} = 0} & (0.23)\end{matrix}$

can be simplified:

0=C ₁ +C ₂ +C ₃

A=−C ₂

B=−C ₃

C ₁ =A+B

Interstitial Concentration C_(I)

$\begin{matrix}{\mspace{79mu} {c_{I} = {{\left( {A + B} \right) \cdot e^{\lambda_{1} \cdot t}} - {A \cdot e^{\lambda_{2} \cdot t}} - {B \cdot e^{\lambda_{3} \cdot t}}}}} & (0.24) \\{\mspace{79mu} {with}} & \; \\{\lambda_{1,2} = {{- \frac{{R_{1} \cdot V_{I}} + {R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}}{2 \cdot V_{P} \cdot V_{I}}} \pm \sqrt{\frac{\left( {{R_{1} \cdot V_{I}} + {R_{2} \cdot V_{I}} + {R_{3} \cdot V_{P}}} \right)^{2}}{4 \cdot V_{P}^{2} \cdot V_{I}^{2}} - \frac{R_{1} \cdot R_{3}}{V_{P} \cdot V_{I}}}}} & (0.25) \\{\mspace{79mu} {\lambda_{3} = {- \frac{R_{4}}{V_{H}}}}} & (0.26) \\{\mspace{79mu} {B = \frac{R_{2} \cdot R_{4} \cdot c_{H,\max}}{\begin{matrix}{{R_{1} \cdot R_{4} \cdot \frac{V_{I}}{V_{H}}} + {R_{2} \cdot R_{4} \cdot \frac{V_{I}}{V_{H}}} +} \\{{R_{3} \cdot R_{4} \cdot \frac{V_{P}}{V_{H}}} - {R_{4}^{2} \cdot \frac{V_{P} \cdot V_{I}}{V_{H}^{2}}} - {R_{1} \cdot R_{3}}}\end{matrix}}}} & (0.27)\end{matrix}$

With 0.8:

$\frac{{dc}_{I}}{dt} = {{c_{P}\frac{R_{2}}{V_{I}}} - {c_{I} \cdot \frac{R_{3}}{V_{I}}}}$c_(P)(0) = 0 c_(I)(0) = 0 $\frac{{dc}_{I}(0)}{dt} = 0$

For the first derivative of eq. 0.24:

$\begin{matrix}{\frac{{dc}_{I}(t)}{dt} = {{\lambda_{1} \cdot \left( {A + B} \right) \cdot e^{\lambda_{1} \cdot t}} - {\lambda_{2} \cdot A \cdot e^{\lambda_{2} \cdot t}} - {\lambda_{3} \cdot B \cdot e^{\lambda_{3} \cdot t}}}} & (0.28) \\{\frac{{dc}_{I}(0)}{dt} = {0 = {{\lambda_{1} \cdot \left( {A + B} \right)} - {\lambda_{2} \cdot A} - {\lambda_{3} \cdot B}}}} & \; \\{A = {B \cdot \frac{\lambda_{1} - \lambda_{3}}{\lambda_{2} - \lambda_{1}}}} & \;\end{matrix}$

Plasma Concentration c_(p)

With eq.0.8 c_(p) is

$\begin{matrix}{\mspace{79mu} {c_{P} = {{\frac{V_{I}}{R_{2}} \cdot \frac{{dc}_{I}}{dt}} + {\frac{R_{3}}{R_{2}} \cdot c_{I}}}}} & (0.29) \\{c_{P} = {{\left( {A + B} \right) \cdot e^{\lambda_{1} \cdot t} \cdot \frac{{V_{I} \cdot \lambda_{1}} + R_{3}}{R_{2}}} - {A \cdot e^{\lambda_{2} \cdot t} \cdot \frac{{V_{I} \cdot \lambda_{2}} + R_{3}}{R_{2}}} - {B \cdot e^{\lambda_{3} \cdot t} \cdot \frac{{V_{I} \cdot \lambda_{3}} + R_{3}}{R_{2}}}}} & \;\end{matrix}$

Another form for eq. 0.29 is:

$\begin{matrix}{c_{P} = {B^{*} \cdot \left\lbrack {{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{1}} + R_{3}} \right) \cdot e^{\lambda_{1} \cdot t}} - {\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right) \cdot e^{\lambda_{2} \cdot t}} - {\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right) \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack}} & (0.30) \\{\mspace{79mu} {with}} & \; \\{\mspace{79mu} {B^{*} = \frac{B}{R_{2} \cdot \left( {\lambda_{2} - \lambda_{1}} \right)}}} & (0.31)\end{matrix}$

Example 3 Four-Compartment Model

A graphical representation of the four-compartment model for renalfunction measurement is shown in FIG. 3A.

Local Interstitiumc_(I;L)

Due to local variations in the skin qualities there are differences insimultaneously measured excretion curves. To address this problem, acompartment ‘Local Interstitium’ is introduced. Two simultaneousmeasurements are used to eliminate the local differences.

The differential equation for c_(I;L) is:

$\begin{matrix}{\frac{{dm}_{I,L}}{dt} = {{{- c_{I,L}} \cdot R_{3,L}} + {c_{P} \cdot R_{2,L}}}} & (0.32) \\{\frac{{dc}_{I,L}}{dt} = {{{- c_{I,L}} \cdot \frac{R_{3,L}}{V_{I,L}}} + {c_{P} \cdot \frac{R_{2,L}}{V_{I,L}}}}} & (0.33)\end{matrix}$

Since the local interstitium is very small compared to the rest of thesystem, the assumption that the plasma marker concentration is notaltered by the tiny mass flow dm_(I;L)/dt is made. So for C_(P) equation0.30 can be used:

$\begin{matrix}{\frac{{dc}_{I,L}}{dt} = {{{- c_{I,L}} \cdot \frac{R_{3,L}}{V_{I,L}}} + {B^{**} \cdot \left\lbrack {{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{1}} + R_{3}} \right) \cdot e^{\lambda_{1} \cdot t}} - {\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right) \cdot e^{\lambda_{2} \cdot t}} - {\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right) \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack}}} & (0.34) \\{\mspace{79mu} {with}} & \; \\{\mspace{79mu} {B^{**} = {B^{*} \cdot \frac{R_{2,L}}{V_{I,L}}}}} & (0.35)\end{matrix}$

Homogenous Solution

$\begin{matrix}{{\frac{{dc}_{I,L}}{dt} + {c_{I,L} \cdot \frac{R_{3,L}}{V_{I,L}}}} = 0} & (0.36) \\{c_{I,L} = {{K(t)} \cdot e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}}} & (0.37)\end{matrix}$

Inhomogeneous Solution

$\begin{matrix}{{g(t)} = {B^{**} \cdot \left\lbrack {{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{1}} + R_{3}} \right) \cdot e^{\lambda_{1} \cdot t}} - {\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right) \cdot e^{\lambda_{2} \cdot t}} - {\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right) \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack}} & (0.38) \\{\mspace{79mu} {{K(t)} = {{\int{{g(t)} \cdot e^{\frac{R_{3,L}}{V_{I,L}} \cdot t} \cdot {dt}}} + G}}} & (0.39) \\{{K(t)} = {{B^{**} \cdot \left\lbrack {{\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{1}} + R_{3}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} \cdot e^{{({\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}})} \cdot t}} - {\frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} \cdot e^{{({\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}})} \cdot t}} - {\frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}} \cdot e^{{({\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}})} \cdot t}}} \right\rbrack} + G}} & (0.40)\end{matrix}$

General Solution

$\begin{matrix}{{c_{I,L}(t)} = {{B^{**} \cdot \left\lbrack {{\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {V_{I} \cdot \lambda_{1} \cdot R_{3}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} \cdot e^{\lambda_{1} \cdot t}} - {\frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} \cdot e^{\lambda_{2} \cdot t}} - {\frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}} \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack} + {G \cdot e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}}}} & (0.41)\end{matrix}$

With c_(I;L)(0)=0

$\begin{matrix}{G = {{- B^{**}} \cdot \left\lbrack {\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {V_{I} \cdot \lambda_{1} \cdot R_{3}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} - \frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} - \frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}}} \right\rbrack}} & (0.42)\end{matrix}$

leading to:

$\begin{matrix}{{c_{I,L}(t)} = {B^{**} \cdot \left\lbrack {{\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {V_{I} \cdot \lambda_{1} \cdot R_{3}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{1} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{2} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{3} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)}} \right\rbrack}} & (0.43)\end{matrix}$

Measured Signal c_(M)

The measured signal is a combined signal from the localinterstitiumc_(I;L) and a very diminished signal from the plasma(vessels near the sensor). The plasma signal is strongly diminishedthrough absorption of the green wavelength in the hemoglobin.

c _(M)=α·(c _(I;L) +χ·c _(P))  (0.44)

αϵ

amplification factor due to electronic componentsχϵ

diminishing factor 0≤χ≤1

fac₁ =χ·α

fac₂=α

for the measured curve c_(M):

$\begin{matrix}{\mspace{79mu} {{c_{M}(t)} = {{{fac}_{1} \cdot {c_{P}(t)}} + {{fac}_{2} \cdot {c_{I,L}(t)}}}}} & (0.45) \\{\mspace{79mu} {with}} & \; \\{\mspace{79mu} {D = {B^{*} \cdot {fac}_{1}}}} & (0.46) \\{\mspace{79mu} {E = {B^{**} \cdot {fac}_{2}}}} & (0.47) \\{{c_{M}(t)} = {{D \cdot \left\lbrack {{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{1}} + R_{3}} \right) \cdot e^{\lambda_{1} \cdot t}} - {\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right) \cdot e^{\lambda_{2} \cdot t}} - {\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right) \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack} + {E \cdot \left\lbrack {{\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{1}} + R_{3}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{1} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {{V_{I} \cdot \lambda_{2}} + R_{3}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{2} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {{V_{I} \cdot \lambda_{3}} + R_{3}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{3} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)}} \right\rbrack}}} & (0.48)\end{matrix}$

Simplification: R₂=R₃

For the diffusion between plasma and interstitium the diffusionconstants forth and backwards should be equal:

R ₂ =R ₃ =R ₂₃   (0.49)

thus, equation 0.25 simplifies to:

$\begin{matrix}{\lambda_{1,2} = {{- \frac{{R_{1} \cdot V_{I}} + {R_{23} \cdot \left( {V_{I} + V_{P}} \right)}}{2 \cdot V_{P} \cdot V_{I}}} \pm \sqrt{\frac{\left( {{R_{1} \cdot V_{I}} + {R_{23} \cdot \left( {V_{I} + V_{P}} \right)}} \right)^{2}}{4 \cdot V_{P}^{2} \cdot V_{I}^{2}} - \frac{R_{1} \cdot R_{23}}{V_{P} \cdot V_{I}}}}} & (0.50) \\{\lambda_{1,2} = {{{- \frac{1}{2}}\left( {\frac{R_{1}}{V_{P}} + \frac{R_{23}}{V_{P}} + \frac{R_{23}}{V_{I}}} \right)} \pm \sqrt{{\frac{1}{4}\left( {\frac{R_{1}}{V_{P}} + \frac{R_{23}}{V_{P}} + \frac{R_{23}}{V_{I}}} \right)^{2}} - {\frac{R_{1}}{V_{P}} \cdot \frac{R_{23}}{V_{I}}}}}} & (0.51) \\{\mspace{79mu} {{\lambda_{1} \cdot \lambda_{2}} = {\frac{R_{1}}{V_{P}} \cdot \frac{R_{23}}{V_{I}}}}} & (0.52) \\{\mspace{79mu} {{\lambda_{1} + \lambda_{2}} = {- \left( {\frac{R_{1}}{V_{P}} + \frac{R_{23}}{V_{P}} + \frac{R_{23}}{V_{I}}} \right)}}} & (0.53)\end{matrix}$

Equation 0.52 can be transformed to:

$\begin{matrix}{R_{23} = \frac{V_{I} \cdot V_{P} \cdot \lambda_{1} \cdot \lambda_{2}}{R_{1}}} & (0.54)\end{matrix}$

Fit Function

Equation 0.54 can be used to replace R₂₃ in equation 0.48:

${c_{M}(t)} = {{D \cdot \left\lbrack {{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \lambda_{1} \cdot V_{I} \cdot \left( {1 + {\frac{V_{P}}{R_{1}} \cdot \lambda_{2}}} \right) \cdot e^{\lambda_{1} \cdot t}} - {\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \lambda_{2} \cdot V_{I} \cdot \left( {1 + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1}}} \right) \cdot e^{\lambda_{2} \cdot t}} - {\left( {\lambda_{2} - \lambda_{1}} \right) \cdot V_{I} \cdot \left( {\lambda_{3} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right) \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack} + {E \cdot \left\lbrack {{\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \lambda_{1} \cdot V_{I} \cdot \left( {1 + {\frac{V_{P}}{R_{1}} \cdot \lambda_{2}}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{1} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \lambda_{2} \cdot V_{I} \cdot \left( {1 + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1}}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{2} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot V_{I} \cdot \left( {\lambda_{3} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{3} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)}} \right\rbrack}}$

which can be simplified with:

$\begin{matrix}{\mspace{79mu} {D^{*} = {D \cdot V_{I}}}} & (0.55) \\{\mspace{79mu} {E^{*} = {E \cdot V_{I}}}} & (0.56) \\{\mspace{79mu} {to}} & \; \\{{c_{M}(t)} = {{D^{*} \cdot \left\lbrack {{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {\lambda_{1} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right) \cdot e^{\lambda_{1} \cdot t}} - {\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {\lambda_{2} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right) \cdot e^{\lambda_{2} \cdot t}} - {\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {\lambda_{3} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right) \cdot e^{\lambda_{3} \cdot t}}} \right\rbrack} + {E^{*} \cdot \left\lbrack {{\frac{\left( {\lambda_{2} - \lambda_{3}} \right) \cdot \left( {\lambda_{1} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right)}{\lambda_{1} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{1} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{1} - \lambda_{3}} \right) \cdot \left( {\lambda_{2} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right)}{\lambda_{2} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{2} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)} - {\frac{\left( {\lambda_{2} - \lambda_{1}} \right) \cdot \left( {\lambda_{3} + {\frac{V_{P}}{R_{1}} \cdot \lambda_{1} \cdot \lambda_{2}}} \right)}{\lambda_{3} + \frac{R_{3,L}}{V_{I,L}}} \cdot \left( {e^{\lambda_{3} \cdot t} - e^{{- \frac{R_{3,L}}{V_{I,L}}} \cdot t}} \right)}} \right\rbrack}}} & (0.57)\end{matrix}$

Function 0.57 is used to fit data of two simultaneous measurements withparameters being either global (same value for both curves) or local—seeTable 1:

TABLE 1 Global and local parameters of fit function 0.57 Parameterglobal local Range λ_(1,3) ✓ <0 $\frac{V_{P}}{R_{1}}$ ✓ >0$\frac{R_{3,L}}{V_{I,L}}$ ✓ >0 D* ✓ >0 E* ✓ >0

In function 0.57 only quotients of R and V can be directly determined.When the quotient of V_(p)/R1 is determined, the plasma volume V_(p) isneeded to get to R1 . Vp becomes an input parameter and can bedetermined by an independent experiment or from literature values, e.g.Probst et al., (2006), Journal of the American Association forLaboratory Animal Science: J A ALAS, 45(2): 49.

$\begin{matrix}{\frac{{dm}_{P}}{dt} = {{{- R_{1}} \cdot c_{P}} + {R_{23} \cdot \left( {c_{I} - c_{P}} \right)} + {\overset{.}{m}}_{in}}} & (0.58) \\{\frac{{dm}_{I}}{dt} = {R_{23} \cdot \left( {c_{P} - c_{I}} \right)}} & (0.59)\end{matrix}$

can be expressed as

$\begin{matrix}{\frac{{dc}_{P}}{dt} = {{{- c_{P}} \cdot \frac{R_{1}}{V_{P}}} + {\left( {c_{I} - c_{P}} \right) \cdot \frac{R_{23}}{V_{P}}} + \frac{{\overset{.}{m}}_{in}}{V_{P}}}} & (0.60) \\{\frac{{dc}_{I}}{dt} = {\frac{R_{23}}{V_{I}} \cdot \left( {c_{P} - c_{I}} \right)}} & (0.61)\end{matrix}$

yields the following differential equation:

$\begin{matrix}{{\frac{d^{2}c_{I}}{{dt}^{2}} + {\frac{{dc}_{I}}{dt} \cdot \left( {\frac{R_{23}}{V_{I}} + \frac{R_{1}}{V_{P}} + \frac{R_{23}}{V_{P}}} \right)} + {c_{I} \cdot \frac{R_{1} \cdot R_{23}}{V_{I} \cdot V_{P}}}} = {{\overset{.}{m}}_{in} \cdot \frac{R_{23}}{V_{I} \cdot V_{P}}}} & (0.62)\end{matrix}$

For c_(I) and c_(p):

$\begin{matrix}{{c_{I}(t)} = {{A \cdot e^{\lambda_{1} \cdot t}} + {B \cdot e^{\lambda_{2} \cdot t}} + \frac{{\overset{.}{m}}_{in}}{R_{1}}}} & (0.63) \\{{c_{p}(t)} = {{A \cdot e^{\lambda_{1} \cdot t} \cdot \left( {1 + \frac{V_{I} \cdot \lambda_{1}}{R_{23}}} \right)} + {B \cdot e^{\lambda_{2} \cdot t} \cdot \left( {1 + \frac{V_{I} \cdot \lambda_{2}}{R_{23}}} \right)} + \frac{{\overset{.}{m}}_{in}}{R_{1}}}} & (0.64)\end{matrix}$

For the steady state:

$\begin{matrix}{c_{P,s} = {{\lim\limits_{t\rightarrow\infty}c_{P}} = \frac{{\overset{.}{m}}_{in}}{R_{1}}}} & (0.65)\end{matrix}$

Relation between R₁ and GFR

$\begin{matrix}{{GFR} = \frac{V_{{glom} \cdot {Filtrat}}}{t}} & (0.66) \\{{\overset{.}{m}}_{{Marker},{{glom} \cdot {Filtrat}}} = {\overset{.}{m}}_{{Marker},{Harn}}} & (0.67) \\{{\overset{.}{m}}_{{Marker},{Harn}} = {\overset{.}{m}}_{{Marker},{in}}} & (0.68) \\{c_{{Marker},{{glom} \cdot {Filtrat}}} = c_{{Marker},{Plasma}}} & (0.69) \\{\frac{c_{{Marker},{Plasma}} \cdot V_{{glom} \cdot {Filtrat}}}{t} = \frac{c_{{Marker},{Harn}} \cdot V_{Harn}}{t}} & (0.70) \\{{GFR} = \frac{c_{{Marker},{Harn}} \cdot V_{Harn}}{t \cdot c_{{Marker},{Plasma}}}} & (0.71) \\{{GFR} = \frac{{\overset{.}{m}}_{{Marker},{in}}}{c_{{Marker},{Plasma}}}} & (0.72) \\{R_{1} = \frac{{\overset{.}{m}}_{{Marker},{in}}}{c_{{Marker},{Plasma}}}} & (0.73) \\{R_{1} = {GFR}} & (0.74)\end{matrix}$

Thus, R₁ equals the GFR.

Example 4 Experimental Validation 1 Introduction 1.1 Purpose

The pharmacokinetic model was validated by the following experiment. Thedistribution and excretion kinetic of the marker is determined byfluorescence measurement on the skin after an i.v. bolus injection inlaboratory animals using four NIC-Kidney devices. After 120 min, whenmost of the marker is excreted, a constant infusion with the same markeris started until a steady state is reached. In this state blood samplesare taken to determine the plasma concentration of the marker. Theconstant infusion experiment is accounted as gold standard method forGFR assessment and allows validation of the results of GFR assessment ofthe bolus experiment combined with the novel pharmacokinetic model(Schock-Kusch et al. (2012) Kidney Int. 82(3): 314-20).

1.2 Equipment

-   -   One NIC-Kidney device with radio frequency unit (RF-Device) for        real-time data observation (Mannheim Pharma& Diagnostics GmbH,        Mannheim Germany)

1Three NIC-Kidney devices with internal memory (Mannheim Pharma &Diagnostics GmbH, Mannheim Germany)

Infusion Pump

-   -   Equipment for blood sampling and fluorescence measurement in        plasma as described elsewhere (Kidney International 2011 Jun;        79(1 1): 1254-8).

1.3 Animals

Male Sprague-Dawley(SD) rats with a body weight of 300-350 g. Themeasurements are conducted on awake animals. Number of animals n=6.

1.4 Drugs

-   -   Injection solution bolus: FITC-Sinistrin 40 mg/ml(Mannheim        Pharma& Diagnostics GmbH, Mannheim Germany)    -   Injection solution constant infusion: FITC-Sinistrin 15 mg/ml    -   Bolus dose: 5 mg/100g b.w.    -   Flow rate of pump during constant infusion: 0.01 ml/min    -   Isofluran(Baxter Deutschland GmbH)

2 Protocol 2.1 Treatment

The animals undergo the standard preparations of catheterization andshaving as described by Schock-Kusch et al. (Kidney International 2011Jun; 79(11): 1254-8). To attach the devices the animals areanaesthetized with Isofluran.

After the devices are attached they should be switched on 10 min priorto injection. One blood sample has to be taken before the injection.

2.2 Bolus Part

After the FITC-Sinistrin injection at t=0 the bolus experiment lasts fort=120 min (Kidney International 2011 Jun; 79(11): 1254-8). During thisperiod all 4 devices record data.

2.3 Constant Infusion Part

At t>120min the infusion pump is switched on and the fluorescence levelis monitored via the RF-device. After reaching a plateau 2 blood samplesare taken in a 10 min interval. The FITC-Sinistrin concentrations in the

plasma of these samples are determined by fluorescence measurement asdescribed elsewhere (Nephrol Dial Transplant. 2009 Oct; 24(10):2997-3001).

2.4 Data Processing

From the gained data the GFR is assessed according to the constantinfusion technique (Kidney Int. 2012 Aug; 82(3): 314-20), and fromcombinations of pairs of the four excretion kinetics assessed during thebolus experiment (6 pairs total) using the novel pharmacological model.

2.5 Sample Curve

See FIG. 3

2.6 Results

Fitting pairs of device combinations with eq. (0.57)

Device Combination 1 & 2 1 & 4 1 & 3 2 & 4 2 & 3 4 & 3 GFR/VP [1/min]0.12 0.14 0.11 0.10 0.12 0.13

The animal (healthy SD rat) had a body weight of m=309g. With a plasmavolume per body weight rate of 0.0412 ml/g (R. J. Probst, J. M. Lim, D.N. Bird, G. L. Pole, A. K. Sato, and J. R. Claybaugh. Gender differencesin the blood volume of conscious Sprague-Dawley rats. Journal of theAmerican Association for Laboratory Animal Science: JAALAS, 45(2):49,2006) the plasma volume is VP=12.7 ml.

Device Combination 1 & 2 1 & 4 1 & 3 2 & 4 2 & 3 4 & 3 GFR [ml/min] 1.501.78 1.46 1.30 1.50 1.62

The blood samples taken during the following constant infusionexperiment showed an average marker concentration of 0.093 mg/ml. With amass flow marker into the animal of 0.15 mg/min the GFR results inGFR=1.61 ml/min (see equation (0.65)).

1-31. (canceled)
 32. A method for determining an organ function in asubject in need thereof, the method comprising: administering afluorescent indicator compound to the bloodstream of the subject;measuring transcutaneously a first concentration-time curve in a firstbody fluid at a first position on the subject, wherein said firstconcentration time curve comprises (i) at least one data point collectedprior to administering the fluorescent indicator compound to thebloodstream of the subject as a first background time point, and (ii) atleast five data points collected after administering the fluorescentindicator compound to the subject; measuring transcutaneously a secondconcentration-time curve in a second body fluid at a second position onthe subject, wherein said second concentration time curve comprises (i)at least one data point collected prior to administering the fluorescentindicator compound to the bloodstream of the subject as a secondbackground time point, and (ii) at least five data points collectedafter administering the fluorescent indicator compound to the subject;fitting the first and the second concentration-time curve into a kineticmodel representing two or three diffusion compartments, wherein thediffusion compartments are selected from the group consisting of plasma,interstitial fluid, local interstitium at the position of the measuring,and distribution; and determining the organ function of the subjectbased on the results of fitting the first concentration-time curve andthe second concentration-time curve into the kinetic model; wherein thefirst position and the second position on the body of the subject aredifferent, the first body fluid and the second body fluid of the subjectmay be the same or different, and measuring fluorescence of theindicator compound at the first position and the second position isperformed simultaneously.
 33. The method according to claim 32, whereinadministering is performed orally, intraarterially, intravenously, or asa bolus injection.
 34. The method according to claim 32, wherein thefluorescent indicator compound is only eliminated from the bloodstreamof the patient by glomerular filtration.
 35. The method according toclaim 32, wherein the kinetic model represents two diffusioncompartments.
 36. The method according to claim 32, wherein the kineticmodel represents three diffusion compartments.
 37. The method accordingto claim 32, wherein the method further comprises determining the bloodvolume of the patient.
 38. The method according to claim 32, wherein theorgan function is the glomerular filtration rate (GFR).
 39. The methodof claim 32, wherein at least at least one of the data points for thefirst concentration-time curve and at least one of the data points forthe second concentration-time curve is collected within two hours orless after administering the fluorescent indicator compound to thesubject.
 40. A device configured to implement the method of claim 32,said device comprising: a first sensor for transcutaneously measuringfluorescence of the indicator at the first position; a second sensor fortranscutaneously measuring fluorescence of the indicator at the secondposition; and a data processing unit for fitting the values obtained bythe sensors into the kinetic model; wherein said first sensor and saidsecond sensor are operatively linked in that the data processing unit ofthe device is configured to bring together the results of themeasurements performed by said sensors and fits the data into thekinetic model.
 41. The device of claim 40, further comprising a datastorage unit comprising at least reference values of total plasma volumecorrelated to body mass.
 42. A kit comprising the device of claim 40 andthe fluorescent indicator compound.
 43. The kit of claim 42, wherein theindicator compound comprises a fluorescent low-molecular weight compoundcovalently bound to a hydrophilic compound, wherein the hydrophilic partis selected from the list consisting of oligo- and polysaccharides,oligo- and polyalcohols, and oligo- and polyethers.
 44. A non-transitorycomputer readable medium comprising an executable computer programconfigured to perform the method according to claim 32, while theprogram is being executed on a computer.
 45. A computer or computernetwork comprising at least one processor, wherein the computer orcomputer network is configured to perform the method according to claim32.